Complexity

Complexity / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 4918342 | https://doi.org/10.1155/2020/4918342

Mansour Shrahili, Naif Alotaibi, Devendra Kumar, A. R. Shafay, "Inference on Exponentiated Power Lindley Distribution Based on Order Statistics with Application", Complexity, vol. 2020, Article ID 4918342, 14 pages, 2020. https://doi.org/10.1155/2020/4918342

Inference on Exponentiated Power Lindley Distribution Based on Order Statistics with Application

Academic Editor: Michele Scarpiniti
Received12 Apr 2020
Revised15 Jul 2020
Accepted16 Jul 2020
Published23 Sep 2020

Abstract

Exponentiated power Lindley distribution is proposed as a generalization of some widely well-known distributions such as Lindley, power Lindley, and generalized Lindley distributions. In this paper, the exact explicit expressions for moments of order statistics from the exponentiated power Lindley distribution are derived. By using these relations, the best linear unbiased estimates of the location and scale parameters, based on type-II right-censored sample, are obtained. Next, the mean, variance, and coefficients of skewness and kurtosis of some certain linear functions of order statistics are calculated and then used to derive the approximate confidence interval for the location and scale parameters using the Edgeworth approximation. Finally, some numerical illustrations and two real data applications are presented.

1. Introduction

Ashour and Eltehiwy [1] introduced the exponentiated power Lindley (EPL) distribution as a generalization of two-parameter power Lindley distribution and they studied some mathematical properties of the EPL distribution. They also showed that this distribution provides more flexibility to analyze a complex real data set. Due to its practicality, the EPL distribution can be used for many applications, including accelerated life testing, survival analysis, reliability, biology, and others. The EPL distribution has attractive feature that its hazard rate function could be decreasing, increasing, and decreasing-increasing-decreasing but not constant, depending on the shape parameter. The EPL distribution has a unimodal and right skewed probability density function (PDF) for . Many authors have developed generalization of Lindley distribution. Prominent among these generalizations which we are aware of are generalized Lindley distribution by Nadarajah et al. [2]; power Lindley distribution by Ghitany et al. [3]; generalized inverse Lindley distribution by Sharma et al. [4]; pseudo-Lindley distribution by Nedjar and Zeghdoudi [5]; size-biased gamma Lindley distribution by Beghriche and Zeghdoud [6]. Ashour and Eltehiwy [1] observed that in numerous situations, the EPL distribution provided a better fit than Lindley, power Lindley, generalized Lindley, exponentiated exponential, modified Weibull, and Weibull distributions.

Over the past six decades or so, several authors have shown keen interest in order statistics. These order statistics and their moments are of great significance in many real life applications involving data relating to flood, drought, reliability, engineering, and life testing situation. In recent times, several papers and books have been published on order statistics and their distributional properties. Among them are David and Nagaraja [7]; Arnold et al. [8]; Balakrishnan and Cohen [9]; Balakrishnan and Ahsanullah [10]; Balakrishnan and Sultan [11]; Malik et al. [12]; Mahmoud et al. [13]; Genc [14]; MirMostafaee [15]; Balakrishnan et al. [16]; Kumar and Dey [17]; Sultan et al. [18]; Sultan and AL-Thubyani [19]; Kumar et al. [20]; and so on. Due to the wide applications of order statistics, several authors have carried out extensive studies on these kinds of ordered data. Numerous papers dealing with moments and estimation of parameters of different lifetime distributions based on order statistics can be found in literature. For example, Balakrishnan and Cohen [9] discussed the moments and estimation of parameters. Sultan et al. [18] obtained the moments of power function distribution. Sultan and AL-Thubyani [19] developed the higher-order moments and inferential procedure for estimating parameters of Lindley distribution. Ahsanullah and Alzaatreh [21] studied the moments and inferential procedure for log-logistic distribution. Recently, Kumar and Goyal [22, 23] and Kumar et al. [24, 25] developed inferential procedure for power Lindley and generalized Lindley, modified power function, and generalized inverse Lindley distribution.

Let be an independent random variable from the EPL distribution with cumulative distribution function (CDF) and PDF is given, respectively, by

The hazard function of the EPL distribution is given bywhere

Some widely well-known distributions can be obtained as special cases from the EPL distribution. By setting and in both (1) and (2), the CDF and PDF of the generalized Lindley distribution and power Lindley distribution are obtained, respectively, while setting gives the CDF and PDF of Lindley distributions.

To the best of our knowledge, there are no reports on moments and estimation for the EPL distribution using order statistics. In this paper, first is to obtain the exact explicit expressions for moments of order statistics from the EPL distribution, and second is to derive the best linear unbiased estimates (BLUEs) of the location and scale parameters of the EPL distribution based on order statistics.

The rest of this paper is organized as follows: in Section 2, the moments of EPL distribution are derived based on order statistics. The BLUEs of the location and scale parameters are provided in Section 3. Next, these BLUEs are used in Section 4 to obtain the coefficient of skewness and kurtosis of some pivotal quantities. Also, we obtain Edgeworth approximations for the distributions of these pivotal quantities. Simulation study and real data analysis are presented in Sections 5 and 6. Finally, the paper ends with a conclusion in Section 7.

2. Moments of Order Statistics

In this section, the moments of order statistics are established for a given random sample from the EPL distribution.

2.1. Single Moments

In this subsection, the explicit expressions for single moments of order statistics are derived. Let be the order statistics for a given random sample from the EPL distribution, and , , denotes the th order statistics with PDF which is given by (see [7])where and are given by (1) and (2), respectively, and

Theorem 1. For the EPL distribution given in (1), the th moment of the th order statistics , and , is as follows:

Proof. From (5), we haveBy using (1) and (2), one can writehence the result.
Some special cases from equation (7):(1)If , we get(2)If , we getas obtained by Kumar and Goyal [23].(3)If and , we get(4)If and , we get(5)If and , we get(6)If and , we getas obtained by Ashour and Eltehiwy [1].

2.2. Double Moments

Let be the order statistics from the EPL distribution. Then, the joint PDF of the th and th order statistics is given aswhere and .

Theorem 2. For the EPL distribution given in (1), the double moment of th and th order statistics, , , is as follows:

Proof. Using (16), we haveBy using the same argument as in Theorem 1, we get the result given in (17).

2.3. Triple Moments

Let denote the order statistics drawn from the EPL distribution. The joint PDF of the th, th, and th order statistics is given bywhere and

Theorem 3. For the EPL distribution given in (1), The triple moment of th, th, and th order statistics , , is as follows:

Proof. From equation (19), we haveBy using the same argument as in Theorems 1 and 2, we obtain the result given in (21).

2.4. Quadruple Moments

Let denote the order statistics drawn from the EPL distribution. The joint PDF of the th, th, th, and th order statistics is given aswhere and

Theorem 4. For the EPL distribution given in (1), the quadruple moment of th, th, th, and th order statistics is as follows:

Proof. From equation (23), we haveBy using the same argument as in Theorem 3, we get the result given in (25).

3. Estimation of Parameters

Here, we study parameter estimation for the EPL distribution based on order statistics.

3.1. BLUEs of Parameters

The PDF of the scale-parameter EPL distribution isand the PDF of the location-scale parameter EPL distribution is

Let , denote type-II right-censored sample from the location-scale parameter EPL distribution in equation (1). Let us denote , , , and , . Therefore,where , and , and . Then, the BLUEs of and can be computed as follows (see Arnold et al. [8]):where

The variances and covariance of these BLUEs can be computed as follows (see Arnold et al. [8]):

The values of and are displayed in Tables 1 and 2 for different values of sample sizes and different censoring cases and for some selected values for . The coefficient of the BLUEs and are given by (17) and (18), respectively, with conditionswhich are used to check the computations accuracy.


,

1701.1367330.005506−0.023661−0.026954−0.024014−0.032131−0.035482
11.161942−0.000791−0.040361−0.029379−0.039803−0.051603
21.215986−0.044508−0.036811−0.061522−0.073146
1001.092681−0.0146520.027777−0.023314−0.001847−0.019939−0.014359−0.014698−0.014324−0.017325
11.105425−0.0051280.000896−0.0176790.001242−0.025909−0.023216−0.012611−0.023022
21.129876−0.005171−0.017773−0.019798−0.008271−0.031966−0.019298−0.027601
31.1367330.005506−0.023616−0.026954−0.024014−0.032131−0.035482
41.161942−0.000796−0.040361−0.029379−0.039803−0.051603

2701.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.038867
11.164716−0.015401−0.017184−0.042061−0.038315−0.051755
21.204871−0.030066−0.038861−0.061012−0.074933
1001.0780950.018066−0.0127110.010341−0.018799−0.013232−0.014746−0.014122−0.014181−0.018722
11.0888180.021605−0.014117−0.007688−0.019094−0.010235−0.016854−0.018684−0.023699
21.1071220.015291−0.019433−0.017916−0.008332−0.019141−0.029435−0.028157
31.1318010.004217−0.030454−0.006347−0.030267−0.030084−0.038867
41.164716−0.015401−0.017184−0.04206−0.038315−0.051755

3701.1259910.003349−0.003324−0.026256−0.035498−0.02695−0.037311
11.158534−0.006501−0.023706−0.036411−0.039839−0.052078
21.203149−0.025636−0.048175−0.053915−0.075387
1001.084513−0.0010820.012616−0.006894−0.0256820.004272−0.023087−0.010631−0.015161−0.018865
11.0896080.0057370.014226−0.023515−0.010205−0.013899−0.022786−0.016716−0.022451
21.113338−0.0069220.010884−0.026283−0.014407−0.026124−0.022595−0.027892
31.1259910.003349−0.003324−0.026256−0.035498−0.026951−0.037311
41.158534−0.006501−0.023706−0.036411−0.039839−0.052078


,

170−0.6071910.0492650.1073570.0985630.1060760.1151310.130798
1−0.6190960.0651740.1482120.1166840.1332730.155753
2−0.6507290.1423760.1601710.1540250.194157
100−0.6153920.0815140.0287930.0381890.0747110.0695230.0770510.0870840.0712460.087283
1−0.6231590.0890410.0272650.0798450.0644780.0872260.0947280.0824620.098115
2−0.6108790.0671920.0426660.1080530.0812350.1026640.0977070.111363
3−0.6071910.0492650.1073570.0985630.1060760.1151310.130798
4−0.6190960.0651740.1482120.1166840.1332730.155753

270−0.6039850.0560410.0865290.0980910.1201220.1170910.126112
1−0.6244020.0817010.1229390.1288540.1404420.150465
2−0.6470630.1458110.1360750.1766380.188541
100−0.5754950.0113320.0709030.0257550.0981210.0403470.0798680.0820630.0819040.085201
1−0.5804270.0105830.0780870.0627710.0814240.0653650.0984930.0876680.096037
2−0.5981870.0436840.0699710.0950370.0771360.0973130.1063560.108691
3−0.6039850.0560410.0865290.0980910.1201220.1170910.126112
4−0.6244020.0817010.1229390.1288540.1404420.150465

370−0.6109730.0714030.0647310.1272950.1173760.0979990.132169
1−0.6355910.1010630.1212870.1327710.1236430.156826
2−0.6352780.1237880.1594740.1571160.194901
100−0.6081310.0678510.0404620.0261150.0872990.0693540.0831180.0793180.0633110.091304
1−0.6014530.0545420.0500850.0453730.0946790.0991950.0820390.0755340.100006
2−0.6050260.0604170.0609510.0638630.1264910.0947470.0851350.113422
3−0.6109730.0714030.0647310.1272950.1173760.0979990.132169
4−0.6355910.1010630.1212870.1327710.1236430.156826

4. Approximate Inference

Here, we derive the confidence intervals for the parameters and based on the following pivotal quantities:where and are the BLUEs of and with variances and , respectively. is used to draw inference for when is known, while can be used to draw inference for when is unknown. Similarly, can be used to draw inference for when is unknown.

The moments presented in Section 2 are used to derive the confidence intervals of the location and scale parameters based on the pivotal quantities in equation (35).

Hence, and can be rewritten aswhere , , is the standardized form of the available type-II right-censored sample , .

we consider finding the approximate distribution by using Edgeworth approximation for a statistic S (with mean 0 and variance 1) aswhere and are the coefficients of skewness and kurtosis of , respectively, and and are the CDF and PDF of the standard normal distribution, respectively.

5. Numerical Aspects

The relations obtained in the preceding sections allow us to evaluate coefficients of the BLUEs. The coefficients of the BLUEs are presented in Tables 1 and 2 while the variances and covariances of the BLUEs are presented in Table 3. From the results presented in Table 3, we observe that the variance of the BLUEs increases as the censoring level increases while the variance of the BLUEs decreases when the sample size increases and increases as increases. In addition, we observe that the covariances of the BLUEs decrease as the censoring level increases while the covariances of the BLUEs increase when the sample size increases and decrease as increases.


Var ()Var ()Cov ()

1700.0688210.135051−0.041267
10.0933330.160655−0.055683
20.1416130.201843−0.083785
1000.0339130.088141−0.020113
10.0420590.100032−0.024623
20.0534050.116162−0.031974
30.0688210.135051−0.041267
40.0933330.160655−0.055683

2700.0705620.135301−0.041106
10.0983880.164369−0.057362
20.1467010.208599−0.085694
1000.0349450.089757−0.020554
10.0423930.101301−0.025082
20.0540410.114633−0.030894
30.0705620.135301−0.041106
40.0983880.164369−0.057362

3700.0720620.137961−0.041643
10.1008750.167103−0.058229
20.1427070.211536−0.084917
1000.0338310.091732−0.020271
10.0421780.103774−0.025141
20.0535430.119151−0.031737
30.0720620.137961−0.041643
40.1008750.167103−0.058229

Tables 46 display the simulated percentage points at and sample size . Table 7 presents the values of the mean, variance, and coefficient of skewness and kurtosis and . The simulated average widths of confidence intervals are presented in Table 8. We observe that the Edgeworth approximations of the distributions of and both work quite satisfactorily; this is also clear from the average width of the confidence intervals based on and which are presented in Table 8.


1%2.50%5%10%90%95%97.50%99%

170−3.686292−3.686298−3.68631−1.234091.4488491.8113242.1909223.049235
1−1.754612−1.459681−1.25981−1.035611.3076551.9249632.4997153.159464
2−1.794954−1.508401−1.28934−1.051821.2984121.8951392.5055793.260451
100−1.483964−1.326313−1.17203−0.997611.3492271.9389662.5498513.414044
1−1.525359−1.336776−1.18298−0.99781.3430921.9145712.5402993.359295
2−1.586863−1.376637−1.20636−1.014231.3381381.9224572.5203713.232563
3−1.630238−1.425475−1.23309−1.017371.3110071.9172572.5171793.225972
4−1.754612−1.459681−1.25981−1.035611.3076551.9249632.4997153.159464

270−1.615403−1.391141−1.21902−1.016191.3199721.9226832.4914233.252833
1−1.663496−1.448206−1.25977−1.032231.3328361.9184222.4949363.207337
2−1.802659−1.515264−1.29324−1.055871.3190811.9074912.4483893.115657
100−1.479384−1.299185−1.15835−0.984511.3436121.9840782.5482373.182261
1−1.505824−1.348861−1.17004−0.995021.3359911.9552582.5337813.205181
2−1.548062−1.357995−1.18398−1.000521.3460911.9505722.4960463.253751
3−1.615403−1.391141−1.21902−1.016191.3199721.9226832.4914233.252833
4−1.663496−1.448206−1.25977−1.032231.3328361.9184222.4949363.207337

370−1.638623−1.422331−1.23184−1.012661.2807851.9039942.5159013.268596
1−1.709657−1.464361−1.24497−1.018471.2769361.9239782.4995093.192122
2−1.805743−1.519798−1.30221−1.052181.2996221.9094212.4571193.214435
100−1.519226−1.329914−1.17265−1.000411.3240021.9371412.5412513.326594
1−1.550921−1.350977−1.18953−1.006241.2937521.9122622.5548413.373878
2−1.604826−1.395428−1.21302−1.01881.2946121.9215912.5120033.376578
3−1.638623−1.422331−1.23184−1.012661.2807851.9039942.5159013.268596
4−1.709657−1.464361−1.24497−1.018471.2769361.9239782.4995093.192122


1%2.50%5%10%90%95%97.50%99%

170−2.678889−2.471173−2.301446−2.0532130.4543720.9571411.4219791.948598
1−2.554143−2.375446−2.212094−1.9747060.5264511.0176251.5042912.077757
2−2.395787−2.253001−2.077632−1.8734220.6305711.1168811.6124112.259385
100−3.019991−2.759098−2.548864−2.2881650.2371740.7209491.0859171.604703
1−2.896556−2.664619−2.453476−2.1990370.3150510.8006661.1871911.704299
2−2.807636−2.583198−2.374634−2.1205140.3927160.8534611.2992171.812966
3−2.678889−2.471173−2.301446−2.0532130.4543720.9571411.4219791.948598
4−2.554143−2.375446−2.212094−1.9747060.5264511.0176251.5042912.077757

270−2.710056−2.505164−2.309474−2.0706790.4486020.9497311.3935451.913112
1−2.582256−2.390288−2.218548−1.9674920.5238741.0320951.5089152.087237
2−2.393321−2.230162−2.073431−1.8626780.6314681.1479241.6060342.213561
100−3.024276−2.730335−2.541782−2.2707120.2751710.7347771.1508981.665428
1−2.898164−2.655736−2.463116−2.2171350.3177730.8015811.2535011.728512
2−2.823059−2.605918−2.400323−2.1435860.3691560.8638721.3250271.825864
3−2.710056−2.505164−2.309474−2.0706790.4486020.9497311.3935451.913112
4−2.582256−2.390288−2.218548−1.9674920.5238741.0320951.5089152.087237

370−2.649813−2.466518−2.258944−2.0413990.4986090.9721351.4377021.937601
1−2.519256−2.344106−2.173079−1.9469060.5682051.0956911.5563112.072295
2−2.371208−2.198918−2.060219−1.8347120.6596111.1760771.6354012.248324
100−2.926982−2.724922−2.529415−2.2550720.2661240.7458791.1682651.687633
1−2.829318−2.645105−2.454084−2.1992880.3463920.8003951.2951531.766573
2−2.748096−2.559353−2.350791−2.1237880.4117250.8955251.3608341.909579
3−2.641983−2.466518−2.258944−2.0413990.4986090.9721351.4377021.937601
4−2.519256−2.344106−2.173079−1.9469060.5682051.0956911.5563112.072295


1%2.50%5%10%90%95%97.50%99%

170−1.291825−1.262013−1.219325−1.1494672.2223953.5238474.9400927.134944
1−1.293906−1.257118−1.216151−1.1405512.3503433.7650885.3893017.852344
2−1.252756−1.219676−1.181444−1.1062552.5459324.1591476.0760958.899025
100−1.323036−1.286828−1.242329−1.1653812.1100513.2236124.4983126.167443
1−1.308061−1.270312−1.231131−1.1616112.1489753.2627314.5249376.631667
2−1.294854−1.264969−1.227509−1.1574872.2529173.4117044.7047526.501364
3−1.291825−1.262013−1.219325−1.1494672.2223953.5238474.9400927.134944
4−1.293906−1.257118−1.216151−1.1405512.3503433.7650885.3893017.852344

270−1.299635−1.254525−1.210859−1.1400812.3200863.5840045.1717397.060132
1−1.269125−1.231214−1.188392−1.1193882.4772813.8719415.4052747.898144
2−1.234267−1.201231−1.165376−1.0941192.6208754.1256885.9525718.910766
100−1.316707−1.279803−1.234296−1.1556442.1277523.2572334.5117555.975934
1−1.318328−1.274231−1.231598−1.1525252.1589893.3807084.5544986.349453
2−1.306309−1.266361−1.220428−1.1427382.2270793.4869144.6948816.876191
3−1.299635−1.254525−1.210859−1.1400812.3200863.5840045.1717397.060132
4−1.269125−1.231214−1.188392−1.1193882.4772813.8719415.4052747.898144

370−1.289032−1.249364−1.210882−1.1316582.2407143.4579874.9177267.291664
1−1.258874−1.224641−1.180973−1.1030542.3198613.7130395.3049178.216081
2−1.243127−1.210756−1.171678−1.0994422.5213284.0719256.0716649.205739
100−1.340722−1.297027−1.250248−1.1740112.0807143.2467394.4594086.277229
1−1.326197−1.285656−1.237163−1.1553452.0906443.2918854.7507226.487507
2−1.313368−1.278006−1.229561−1.1521972.1731853.3662414.7640536.650645
3−1.289032−1.249364−1.210882−1.1316582.2407143.4579874.9177267.291664
4−1.258874−1.224641−1.180973−1.1030542.3198613.7130395.3049178.216081


MeanMean

17000.0395181.2934152.911714−0.8698120.0775510.7424140.979527
100.0535941.1925612.520105−0.7974930.0922520.7797771.008172
200.0813181.1200992.226028−0.7114870.1159030.8556711.063987
10000.0194741.4233283.072102−1.0766740.0506130.5777310.656743
100.0241521.3905292.929123−1.0106570.0574410.6469920.849696
200.0306661.3138592.779625−0.9378680.0667030.7043260.996806
300.0395181.2934152.911714−0.8698120.0775510.7424140.979527
400.0535941.1925612.520105−0.7974930.0922520.7797771.008172

27000.0404461.3461192.959134−0.8722350.0775540.6573310.621787
100.0563961.2905112.950683−0.7913610.0942150.7116860.724204
200.0840881.1549962.724035−0.7024710.1195680.8457661.076625
10000.0200311.5209313.976296−1.0709040.0514480.5638390.452005
100.0242991.4646893.627171−1.0080420.0580650.5899490.502475
200.0309761.4463713.457844−0.9476090.0657070.6258930.561158
300.0404461.3461192.959134−0.8722350.0775540.6573310.621787
400.0563961.2905112.950683−0.7913610.0942150.7116860.724204

37000.0413361.4061693.581796−0.8624830.0791360.6784380.457074
100.0578641.3383693.284062−0.7836760.0958530.7577210.647221
200.0818591.1676492.806261−0.6965230.1213410.8704860.981312
10000.0194061.4209983.140231−1.0577130.0526190.5634010.372104
100.0241941.4683183.741814−0.9944510.0595260.5901850.356138
200.0307131.3754743.216648−0.9280650.0683470.6318020.412006
300.0413361.4061693.581796−0.8624830.0791360.6784380.457074
400.0578641.3383693.284062−0.7836760.0958530.7577210.647221


90%95%90%95%90%95%

1703.1416983.8825633.2592043.8987093.0186123.606341
13.1781943.9431433.2506423.8992023.1814153.910418
23.2007273.9636543.2213553.8361943.2747093.101615
1003.1424293.8474223.2765593.8812343.4757813.902103
13.1252973.8826413.2646983.9092363.9810263.154185
23.1345473.8540413.2641953.9309443.5603453.925101
33.1416983.8825633.2592043.8987093.6701653.258404
43.1781943.9431433.2506423.8992023.4137243.135169

2703.1503453.9426543.2585853.8931523.4361963.853183
13.1847693.9593943.2297193.8797383.0665343.860473
23.1844774.0139813.1945133.8654113.9496763.866606
1003.1109973.8761643.2698133.8450163.4357063.873179
13.0975453.8770753.2541423.8518813.8468343.109103
23.1288143.8970083.2280953.8824153.4893363.951775
33.1503453.9426543.2585853.8931523.3693413.785608
43.1847693.9593943.2297193.8797383.3139533.148138

3703.1358343.9382313.2310793.9042213.8891423.348875
13.1689483.9638713.2687713.9004173.9719833.771145
23.2116313.9769173.2362953.8343283.0971213.096853
1003.1097863.8711663.2753293.8931873.4198193.801354
13.1017883.9058173.2544813.9402583.8767753.313559
23.1346113.9074313.2463163.9201873.4216463.916297
33.1358343.9382313.2310793.9042213.6015933.065644
43.1689483.9638713.2687713.9004173.2424893.029839

6. Real Data Analysis

In this section, we analyze two real data sets to show the importance of the proposed estimators. One data set from environmental monitoring and another data is from maximum flood level for the Susquehanna River at Harrisburg.

Example 1. Analysis of clean upgradient ground-water monitoring wells subjected to mg/L.In the first data set, we consider vinyl chloride data obtained from clean upgradient monitoring wells which was studied by many authors such as Bhaumik and Gibbons [26]; Krishnamoorthy et al. [27]; Bhaumik et al. [28]; and Kumar and Goyal [22, 23]. The data are as follows:Now, a random sample of size 10 is selected from the given data set 1.0, 1.2, 3.2, 2.4, 0.8, 2.0, 0.4, 0.2, 2.9, 1.2. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate of , , and . By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding values are 0.163772 and 0.9513, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moveover, the empirical cumulative distribution function (ECDF) plot and the Quantile-Quantile (Q-Q) plots are also presented under the vinyl chloride data as shown in Figure 1, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables 1 and 2, we have

Example 2. Analysis of the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania.The second data set presents the maximum flood level for the Susquehanna River at Harrisburg, Pennsylvania, and it was studied by Dumonceaux and Antle [29]. The data areNow, a random sample of size 10 is selected from the given data set 0.418, 0.265, 0.613, 0.269, 0.654, 0.338, 0.315, 0.379, 0.297, 0.412. By the EPL distribution for the given sample, we have obtained the maximum likelihood estimate of , , and . By computation, the Kolmogorov–Smirnov (K–S) distance and the corresponding values are 0.14282 and 0.9694, respectively, which implies that the EPL distribution provides a reasonable model for this data. Moreover, as a further illustration, the empirical cumulative distributions plot and overlay the theoretical EPL distribution, the quantile-quantile (Q-Q) plots are also presented under the maximum flood level data, as shown in Figure 2, which also imply that the EPL distribution can be used as a proper model to fit this data.
Then, by using the BLUE coefficients in Tables 1 and 2, we have

7. Conclusion

In this paper, we have considered the EPL distribution when data are available in the form of order statistics. We first presented expressions for the single, double, triple, and quadruple moments. By using these moments, we have calculated the BLUEs of the location and scale parameters and the coefficient of skewness and kurtosis for some linear pivotal quantities. In the simulation study, we observed that the variance of the BLUEs increases when a high censoring level is taken into account; however, it decreases when a large value of parameter is considered corresponding to a fixed value of parameters and . Also, the covariance of the BLUEs decreases as the censoring level increases while it increases when the sample size increases and it decreases as increases. We next considered the distributions of the pivotal quantities in terms of Edgeworth approximation. These pivotal quantities are used to construct the interval estimation for the location and scale parameters. From our finding, we see that the moments of order statistics of the distribution are well behaved. This will encourage the study of the other properties of order statistics for a future research.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG-1438-086.

References

  1. S. K. Ashour and M. A. Eltehiwy, “Exponentiated power Lindley distribution,” Journal of Advanced Research, vol. 6, no. 6, pp. 895–905, 2015. View at: Publisher Site | Google Scholar
  2. S. Nadarajah, H. S. Bakouch, and R. Tahmasbi, “A generalized lindley distribution,” Sankhya B, vol. 73, no. 2, pp. 331–359, 2011. View at: Publisher Site | Google Scholar
  3. M. E. Ghitany, D. K. Al-Mutairi, N. Balakrishnan, and L. J. Al-Enezi, “Power Lindley distribution and associated inference,” Computational Statistics & Data Analysis, vol. 64, pp. 20–33, 2013. View at: Publisher Site | Google Scholar
  4. V. K. Sharma, S. K. Singh, U. Singh, and F. Merovci, “The generalized inverse lindley distribution: a new inverse statistical model for the study of upside-down bathtub data,” Communications in Statistics-Theory and Methods, vol. 45, no. 19, pp. 5709–5729, 2016. View at: Publisher Site | Google Scholar
  5. S. Nedjar and H. Zeghdoudi, “On Pseudo lindley distribution: properties and applications,” New Trends in Mathematical Science, vol. 1, no. 5, pp. 59–65, 2017. View at: Publisher Site | Google Scholar
  6. A. Beghriche and H. Zeghdoud, “A size biased gamma lindley distribution,” Thailand Statistician, vol. 17, pp. 179–189, 2019. View at: Google Scholar
  7. H. A. David and H. N. Nagaraja, Order Statistics, Wiley, New York, NY, USA, 3rd edition, 2003.
  8. B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics, John Wiley, New York, NY, USA, 2003.
  9. N. Balakrishnan and A. C. Cohen, Order Statistics and Inference: Estimation Methods, Academic Press, San Diego, CA, USA, 1991.
  10. N. Balakrishnan and M. Ahsanullah, “Relations for single and product moments of record values from exponential distribution,” Journal of Applied Statistical Science, vol. 2, pp. 73–87, 1995. View at: Google Scholar
  11. N. Balakrishnan and K. S. Sultan, “Recurrence relations and identities for moments of order statistics,” in Handbook of Statistics, N. Balakrishnan and C. R. Rao, Eds., vol. 16, pp. 149–228, Elsevier, Amsterdam, Netherlands, 1998. View at: Google Scholar
  12. H. J. Malik, N. Balakrishnan, and S. E. Ahmed, “Recurrence relations and identities for moments of order statistics. I: arbitrary continuous distribution,” Communications in Statistics-Theory and Methods, vol. 17, pp. 2623–2655, 1988. View at: Google Scholar
  13. M. A. W. Mahmoud, K. S. Sultan, and S. M. Amer, “Order statistics from inverse weibull distribution and associated inference,” Computational Statistics & Data Analysis, vol. 42, no. 1-2, pp. 149–163, 2003. View at: Publisher Site | Google Scholar
  14. A. İ. Genç, “Moments of order statistics of topp-leone distribution,” Statistical Papers, vol. 53, no. 1, pp. 117–131, 2012. View at: Publisher Site | Google Scholar
  15. S. M. T. K. MirMostafaee, “On the moments of order statistics coming from the Topp-Leone distribution,” Statistics & Probability Letters, vol. 95, pp. 85–91, 2014. View at: Publisher Site | Google Scholar
  16. N. Balakrishnan, X. Zhu, and B. Al-Zahrani, “Recursive computation of the single and product moments of order statistics from the complementary exponential-geometric distribution,” Journal of Statistical Computation and Simulation, vol. 85, no. 11, pp. 2187–2201, 2015. View at: Publisher Site | Google Scholar
  17. D. Kumar and S. Dey, “Power generalized Weibull distribution based on order statistics,” Journal of Statistical Research, vol. 51, pp. 61–78, 2017. View at: Google Scholar
  18. K. S. Sultan, A. Childs, and N. Balakrishnan, “Higher order moments of order statistics from the power function distribution and Edgeworth approximate inference,” in Advances in Stochastic Simulation and Methods, N. Balakrishnan, V. B. Melas, and S. Ermakove, Eds., pp. 245–282, Birkhauser, Boston, MA, USA, 2000. View at: Google Scholar
  19. K. S. Sultan and W. S. AL-Thubyani, “Higher order moments of order statistics from the Lindley distribution and associated inference,” Journal of Statistical Computation and Simulation, vol. 86, no. 17, pp. 3432–3445, 2016. View at: Publisher Site | Google Scholar
  20. D. Kumar, S. Dey, and S. Nadarajah, “Extended exponential distribution based on order statistics,” Communications in Statistics-Theory and Methods, vol. 46, no. 18, pp. 9166–9184, 2017. View at: Publisher Site | Google Scholar
  21. M. Ahsanullah and A. Alzaatreh, “Parameter estimation for the log-logistic distribution based on order statistics,” REVSTAT Statistical Journal, vol. 16, pp. 429–443, 2018. View at: Google Scholar
  22. D. Kumar and A. Goyal, “Order Statistics from the power Lindley distribution and associated inference with application,” Annals of Data Science, vol. 6, no. 1, pp. 153–177, 2019. View at: Publisher Site | Google Scholar
  23. D. Kumar and A. Goyal, “Generalized lindley distribution based on order statistics and associated inference with application,” Annals of Data Science, vol. 6, no. 4, pp. 707–736, 2019. View at: Publisher Site | Google Scholar
  24. D. Kumar, M. Kumar, and J. P. S. Joorel, “Estimation with modified power function distribution based on order statistics with application to evaporation data,” Annals of Data Science, 2020. View at: Publisher Site | Google Scholar
  25. D. Kumar, M. Nassar, and S. Dey, “Inference for generalized inverse Lindley distribution based on generalized order statistics,” Afrika Matematika, 2020. View at: Publisher Site | Google Scholar
  26. D. K. Bhaumik and R. D. Gibbons, “One-sided approximate prediction intervals for at Leastpofm observations from a gamma population at each ofrLocations,” Technometrics, vol. 48, no. 1, pp. 112–119, 2006. View at: Publisher Site | Google Scholar
  27. K. Krishnamoorthy, T. Mathew, and S. Mukherjee, “Normal-based methods for a gamma distribution,” Technometrics, vol. 50, no. 1, pp. 69–78, 2008. View at: Publisher Site | Google Scholar
  28. D. K. Bhaumik, K. Kapur, and R. D. Gibbons, “Testing parameters of a gamma distribution for small samples,” Technometrics, vol. 51, no. 3, pp. 326–334, 2009. View at: Publisher Site | Google Scholar
  29. R. Dumonceaux and C. E. Antle, “Discrimination between the log-normal and the weibull distributions,” Technometrics, vol. 15, no. 4, pp. 923–926, 1973. View at: Publisher Site | Google Scholar

Copyright © 2020 Mansour Shrahili et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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